It's already been discussed if Trump can withdraw from the Paris agreement. In one of the answers @Brythan mentions the following:
In order to be binding in the United States (US), a treaty must be ratified by two thirds of the Senate (67 Senators). UN COP21 was not. It was just signed by Barack Obama.
Does this mean that the Paris Agreement was never legally binding in the first place? If so, what's the big deal about Trump "leaving" it?
For a treaty to be binding, it needs to be ratified by the Senate. Article II, Section 2, paragraph 2 of the United States Constitution states:
He shall have power, by and with the advice and consent of the Senate, to make treaties, provided two thirds of the Senators present concur;
Since former President Obama signed it by unilateral executive action, it is only binding to his administration. Thus, it is not a treaty and instead an executive agreement for the United States.
This article by The Hill also explains this:
The Administration argues that the Accord is not a treaty but rather an executive agreement between President Obama and other nations. Even so, the President cannot bind the country with an executive agreement; he can only bind his administration.
Thus, the Paris Agreement is either an unratified treaty—in which case it has no effect—or it is an agreement only with the Obama Administration—in which case it is only valid for the nine months until his administration ends. Either way, the agreement is ineffective come January.
As for why it's a big deal about Trump withdrawing from it, it basically makes it official that the US is not part of the agreement anymore.
Though the article above mentions that it's ineffective once Obama's administration, it's disputed as the US has deposited their instruments of acceptance with the Depositary. Currently, the US still has the status of having ratified the agreement. So, Trump's action makes the withdrawal official.
Also, it's worth noting that the commitments and outcome achieved by each country is not legally binding:
What obligations do countries have under the agreement to reduce their emissions?
The Paris Agreement establishes a set of binding procedural commitments. Parties commit to “prepare, communicate and maintain” successive NDCs; to “pursue domestic mitigation measures” aimed at achieving their NDCs; and to regularly report on their emissions and on progress in implementing their NDCs. The agreement also sets the expectation that each party’s successive NDC will “represent a progression” beyond its previous one and “reflect its highest possible ambition.” The achievement by a party of its NDCs is not a legally binding obligation.
(emphasis mine)
My other answer goes into more detail regarding the impact of staying in or leaving the agreement.
The Paris agreement does not have any legal power anywhere. It is and always has been a framework for voluntary cooperation between nations to reduce harmful climate change. The only thing required of participant nations was for them to submit a plan, called a "Nationally Determined Contribution" or NDC, for what they thought they could do over the next few decades to reduce greenhouse gas emissions and to make a statement that they would give their best efforts to meet the targets in their NDC. That's it. Actually meeting the NDC targets? Voluntary. Contributions to the Green Climate Fund? Also voluntary. Thus, whether the agreement ever got ratified in the Senate was irrelevant because the agreement didn't actually obligate the US to do anything.
Trump leaving the agreement isn't really that big a deal, at least in practical terms. For one thing, a subsequent President could just as easily rejoin the agreement, and four years' delay isn't that long in the grand scheme of emissions mitigation. For another thing, much of the US NDC was based on observations of ongoing trends in the US energy markets. There is a lot of inertia in those markets, and it's really not clear that there is anything Trump can do to stop the shift from coal to gas, or the falling prices of renewables. If large states like California decide to embark on their own mitigation initiatives on top of that, then the US might not stray too far from its NDC projections, despite withdrawing from the agreement.
In political terms, the effect of Trump's announcement is to communicate the President's attitudes toward the whole subject of climate change. Predictably, his base loves it, and his opposition hates it. It's hard to imagine that the announcement really changed anybody's mind about him. He's made his opinions on the subject pretty clear already. Probably the best thing the announcement can do, from Trump's point of view, is to supplant other issues in the news that he'd rather not talk about. In this respect, the announcement has been reasonably effective.
Update: After exchanging some comments from the author of the question, it seems that the intent of the question was to ask about the role of Senate ratification in an arrangement like the Paris agreement. The short answer is that Senate action or lack thereof is mostly irrelevant to the status of the Paris agreement. For more detail, read on.
The first thing to understand is that US law actually recognizes three types of international accords:
One source of confusion is that all of these might be referred to as "treaties" in the context of international relations, but only the first counts as a "treaty" in the context of Article II of the US Constitution.
Many people will be surprised to learn that Article II treaties are just a small part of American foreign policy. According to the Wikipedia link above (which itself draws on Treaties and other International Agreements: the Role of the United States Senate, a report published by the Congressional Research Service):
Between 1946 and 1999, the United States completed nearly 16,000 international agreements. Only 912 of those agreements were treaties, submitted to the Senate for approval as outlined in Article II of the United States Constitution. Since the Franklin Roosevelt presidency, only 6% of international accords have been completed as Article II treaties. Most of these executive agreements consist of congressional-executive agreements.
So, where does that leave the Paris agreement? Because the Paris agreement does not make any binding demands on the nations that agree to it, it is trivially within the President's authority. Presidents do not require Congressional approval to announce aspirational goals. Actually attaining those goals might require help from Congress, or it might not, since the US NDC was structured to fit within the regulatory authority the Executive Branch thought it already had. However, these concerns are not particularly relevant to the Paris agreement because it doesn't actually require signatory nations to meet their NDC goals; it only requires them to state those goals, and that is surely within the Executive's authority.
